Applied and Computational Harmonic Analysis最新文献

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The stability of generalized phase retrieval problem over compact groups 紧群上广义相位恢复问题的稳定性

IF 3.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2025-12-11 DOI: 10.1016/j.acha.2025.101838

Tal Amir , Tamir Bendory , Nadav Dym , Dan Edidin

The generalized phase retrieval problem over compact groups aims to recover a set of matrices–representing an unknown signal–from their associated Gram matrices. This framework generalizes the classical phase retrieval problem, which reconstructs a signal from the magnitudes of its Fourier transform, to a richer setting involving non-abelian compact groups. In this broader context, the unknown phases in Fourier space are replaced by unknown orthogonal matrices that arise from the action of a compact group on a finite-dimensional vector space. This problem is primarily motivated by advances in electron microscopy to determining the 3D structure of biological macromolecules from highly noisy observations. To capture realistic assumptions from machine learning and signal processing, we model the signal as belonging to one of several broad structural families: a generic linear subspace, a sparse representation in a generic basis, the output of a generic ReLU neural network, or a generic low-dimensional manifold. Our main result shows that, for a prior of sufficiently low dimension, the generalized phase retrieval problem not only admits a unique solution (up to inherent group symmetries), but also satisfies a bi-Lipschitz property. This implies robustness to both noise and model mismatch—an essential requirement for practical use, especially when measurements are severely corrupted by noise. These findings provide theoretical support for a wide class of scientific problems under modern structural assumptions, and they offer strong foundations for developing robust algorithms in high-noise regimes.

紧群上的广义相位恢复问题旨在从其相关的Gram矩阵中恢复一组表示未知信号的矩阵。该框架推广了经典的相位恢复问题，该问题从其傅里叶变换的幅度重建信号，到涉及非阿贝尔紧群的更丰富的设置。在这个更广泛的背景下，傅里叶空间中的未知相位被未知的正交矩阵所取代，这些正交矩阵是由有限维向量空间上紧群的作用产生的。这个问题的主要动机是电子显微镜技术的进步，可以从高噪声的观察中确定生物大分子的三维结构。为了从机器学习和信号处理中获取现实的假设，我们将信号建模为属于几个广泛结构族之一：一般线性子空间，一般基中的稀疏表示，一般ReLU神经网络的输出，或一般低维流形。我们的主要结果表明，对于足够低维的先验，广义相位恢复问题不仅存在唯一解（不超过固有群对称），而且满足双lipschitz性质。这意味着对噪声和模型不匹配都具有鲁棒性——这是实际使用的基本要求，特别是当测量结果受到噪声的严重破坏时。这些发现为现代结构假设下的广泛科学问题提供了理论支持，并为在高噪声条件下开发健壮的算法提供了坚实的基础。

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引用次数: 0

Assembly and iteration: Transition to linearity of wide neural networks 装配与迭代：广义神经网络向线性的过渡

IF 3.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2025-12-08 DOI: 10.1016/j.acha.2025.101834

Chaoyue Liu , Libin Zhu , Mikhail Belkin

The recently discovered remarkable property that very wide neural networks in certain regimes are linear functions of their weights has become one of the key insights into understanding the mathematical foundations of deep learning. In this work, we show that this transition to linearity of wide neural networks can be viewed as an outcome of an iterated assembly procedure employed in the construction of neural networks. From the perspective of assembly, the output of a wide network can be viewed as an assembly of a large number of similar sub-models, which will transition to linearity as their number increases. This process can be iterated multiple times to show the transition to linearity of deep networks, including general feedforward neural networks with Directed Acyclic Graph (DAG) architecture.

最近发现的一个显著特性是，在某些情况下，非常广泛的神经网络是其权重的线性函数，这已经成为理解深度学习数学基础的关键见解之一。在这项工作中，我们表明这种向广义神经网络线性的过渡可以被视为神经网络构建中采用的迭代组装过程的结果。从装配的角度来看，宽网络的输出可以看作是大量相似子模型的装配，随着子模型数量的增加，子模型会向线性过渡。这个过程可以多次迭代，以显示深度网络向线性的过渡，包括具有有向无环图（DAG）架构的一般前馈神经网络。

引用次数: 0

Parameterized proximal-gradient algorithms for L1/L2 sparse signal recovery L1/L2稀疏信号恢复的参数化近端梯度算法

IF 3.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2025-12-04 DOI: 10.1016/j.acha.2025.101835

Na Zhang , Xinrui Liu , Qia Li

In this paper, we consider a new ℓ₁/ℓ₂ (the ratio of ℓ₁ and ℓ₂ norms) based sparse signal recovery model, which incorporates a ball constraints to ensure the existence of optimal solutions. The presence of two constraints in this model causes algorithmic difficulties for its numerical treatment. To overcome the difficulties, we propose a penalty formulation for the model, and establish the relationships between their optimal solutions and stationary points. Inspired by the parametric approach for fractional programs, we further propose a parameterized proximal-gradient algorithm (PPGA) and its line search counterpart (PPGA

_

L) for solving a general structured fractional program having the penalty problem as a special case. In particular, we derive a closed form solution to the proximal operator of some nonconvex function, which is required to compute in each iteration when specializing the proposed algorithms to the penalty problem. Moreover, we prove the global convergence of the entire sequences generated by PPGA and PPGA

_

L with monotone line search for the penalty problem. Numerical experiments demonstrate the efficiency of the proposed algorithms for noise-free and noisy signal recovery.

本文考虑了一种新的基于l_1 / l_2 （l_1与l_2范数之比）的稀疏信号恢复模型，该模型引入了球约束以保证最优解的存在性。该模型中存在两个约束条件，这给其数值处理带来了算法上的困难。为了克服这个困难，我们提出了模型的惩罚公式，并建立了它们的最优解与平稳点之间的关系。在分数阶规划参数化方法的启发下，我们进一步提出了一种参数化的近端梯度算法（PPGA）及其对应的线搜索算法（PPGA_L），用于求解一类特殊情况下具有惩罚问题的一般结构化分数阶规划。特别地，我们导出了一些非凸函数的近算子的闭形式解，当将所提出的算法专门化到惩罚问题时，需要在每次迭代中计算近算子。此外，我们还利用单调线搜索证明了PPGA和PPGA_L生成的整个序列的全局收敛性。数值实验证明了所提算法在无噪声和有噪声信号恢复方面的有效性。

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引用次数: 0

Theoretical guarantees for low-rank compression of deep neural networks 深度神经网络低秩压缩的理论保证

IF 3.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2025-12-02 DOI: 10.1016/j.acha.2025.101837

Shihao Zhang , Rayan Saab

Deep neural networks have achieved state-of-the-art performance across numerous applications, but their high memory and computational demands present significant challenges, particularly in resource-constrained environments. Model compression techniques, such as low-rank approximation, offer a promising solution by reducing the size and complexity of these networks while only minimally sacrificing accuracy. In this paper, we develop an analytical framework for data-driven post-training low-rank compression. We prove three recovery theorems under progressively weaker assumptions about the approximate low-rank structure of activations, modeling deviations via noise. Our results represent a step toward explaining why data-driven low-rank compression methods outperform data-agnostic approaches and towards theoretically grounded compression algorithms that reduce inference costs while maintaining performance.

深度神经网络已经在许多应用中实现了最先进的性能，但其高内存和计算需求带来了重大挑战，特别是在资源受限的环境中。模型压缩技术，如低秩近似，提供了一个有前途的解决方案，通过减少这些网络的大小和复杂性，同时只牺牲最小的准确性。在本文中，我们开发了一个数据驱动的训练后低秩压缩分析框架。我们在关于激活的近似低秩结构的逐渐变弱的假设下证明了三个恢复定理，通过噪声建模偏差。我们的研究结果向解释为什么数据驱动的低秩压缩方法优于数据不可知的方法，以及在保持性能的同时降低推理成本的理论基础压缩算法迈出了一步。

引用次数: 0

High-order synchrosqueezed chirplet transforms for multicomponent signal analysis 用于多分量信号分析的高阶同步压缩啁啾变换

IF 3.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2025-12-01 DOI: 10.1016/j.acha.2025.101839

Yi-Ju Yen , De-Yan Lu , Sing-Yuan Yeh , Jian-Jiun Ding , Chun-Yen Shen

This study focuses on the analysis of signals containing multiple components with crossover instantaneous frequencies (IF). This problem was initially solved with the chirplet transform (CT). Also, it can be sharpened by adding the synchrosqueezing step, which is called the synchrosqueezed chirplet transform (SCT). However, we found that the SCT goes wrong with the high chirp modulation signal due to the wrong estimation of the IF. In this paper, we present the improvement of the post-transformation of the CT. The main goal of this paper is to amend the estimation introduced in the SCT and carry out the high-order synchrosqueezed chirplet transform. The proposed method reduces the wrong estimation when facing a stronger variety of chirp-modulated multi-component signals. The theoretical analysis of the new reassignment ingredient is provided. Numerical experiments on some synthetic signals are presented to verify the effectiveness of the proposed high-order SCT.

本研究主要针对具有交叉瞬时频率（IF）的多分量信号进行分析。这个问题最初是用小波变换（CT）来解决的。此外，它可以通过增加同步压缩步骤来锐化，这被称为同步压缩啁啾变换（SCT）。然而，我们发现由于对中频的错误估计，SCT在高啁啾调制信号下会出错。在本文中，我们提出了改进后变换的CT。本文的主要目的是对SCT中引入的估计进行修正，并进行高阶同步压缩小波变换。该方法减少了在面对多种啁啾调制多分量信号时的错误估计。对这种新的重分配成分进行了理论分析。在一些合成信号上进行了数值实验，验证了该方法的有效性。

引用次数: 0

Universal approximation property of fully convolutional neural networks with zero padding 具有零填充的全卷积神经网络的普遍逼近性质

IF 3.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2025-11-29 DOI: 10.1016/j.acha.2025.101833

Geonho Hwang , Myungjoo Kang

The Convolutional Neural Network (CNN) is one of the most prominent neural network architectures in deep learning. Despite its widespread adoption, our understanding of its universal approximation properties has been limited due to its intricate nature. CNNs inherently function as tensor-to-tensor mappings, preserving the spatial structure of input data. However, limited research has explored the universal approximation properties of fully convolutional neural networks as arbitrary continuous tensor-to-tensor functions. In this study, we demonstrate that CNNs, when utilizing zero padding, can approximate arbitrary continuous functions in cases where both the input and output values exhibit the same spatial shape. Additionally, we determine the minimum depth of the neural network required for approximation. We also verify that deep, narrow CNNs possess the UAP as tensor-to-tensor functions. The results encompass a wide range of activation functions, and our research covers CNNs of all dimensions.

卷积神经网络（CNN）是深度学习中最突出的神经网络架构之一。尽管它被广泛采用，但由于其复杂的性质，我们对其普遍近似性质的理解受到限制。cnn本质上是张量到张量的映射，保留了输入数据的空间结构。然而，有限的研究探索了全卷积神经网络作为任意连续张量-张量函数的普遍逼近性质。在本研究中，我们证明了当使用零填充时，cnn可以在输入和输出值具有相同空间形状的情况下近似任意连续函数。此外，我们确定了逼近所需的神经网络的最小深度。我们还验证了深度，窄cnn具有UAP作为张量-张量函数。结果包含了广泛的激活函数，我们的研究涵盖了所有维度的cnn。

引用次数: 0

Recovering a group from few orbits 从几个轨道中恢复一个群

IF 3.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2025-11-29 DOI: 10.1016/j.acha.2025.101836

Dustin G. Mixon , Brantley Vose

For an unknown finite group G of automorphisms of a finite-dimensional Hilbert space, we find sharp bounds on the number of generic G-orbits needed to recover G up to group isomorphism, as well as the number needed to recover G as a concrete set of automorphisms.

对于有限维希尔伯特空间中未知的有限自同构群G，我们找到了使G恢复到群同构所需的一般G轨道的数目，以及使G恢复为一个具体的自同构集所需的数目的明显界限。

引用次数: 0

Painless construction of unconditional bases for anisotropic modulation and Triebel-Lizorkin type spaces 各向异性调制和triiebel - lizorkin型空间无条件基的无痛构造

IF 3.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2025-11-28 DOI: 10.1016/j.acha.2025.101832

Morten Nielsen

We construct smooth localized orthonormal bases compatible with anisotropic Triebel-Lizorkin and Besov type spaces on

R^{d}

. The construction is based on tensor products of so-called univariate brushlet functions that are based on local trigonometric bases in the frequency domain, and the construction is painless in the sense that all parameters for the construction are explicitly specified. It is shown that the associated decomposition system form unconditional bases for the full family of Triebel-Lizorkin and Besov type spaces, including for the so-called α-modulation and α-Triebel-Lizorkin spaces. In the second part of the paper we study nonlinear m-term approximation with the constructed bases, where direct Jackson and Bernstein inequalities for m-term approximation with the tensor brushlet system in α-modulation and α-Triebel-Lizorkin spaces are derived. The inverse Bernstein estimates rely heavily on the fact that the constructed system is non-redundant.

我们在Rd上构造与各向异性triiebel - lizorkin和Besov型空间兼容的光滑局域正交基。该构造基于所谓的单变量刷波函数的张量积，该函数基于频域的局部三角基，并且构造的所有参数都明确指定，因此构造是无痛的。结果表明，对于triiebel - lizorkin和Besov型空间，包括α-调制和α- triiebel - lizorkin空间，相关分解体系形成了无条件基。在本文的第二部分，我们用所构造的基研究了非线性m项逼近，导出了在α-调制和α- triiebel - lizorkin空间中张量刷波系统的m项逼近的直接Jackson和Bernstein不等式。逆伯恩斯坦估计在很大程度上依赖于所构建的系统是非冗余的这一事实。

引用次数: 0

Empirical plunge profiles of time-frequency localization operators 时频定位算子的经验波动曲线

IF 3.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2025-11-15 DOI: 10.1016/j.acha.2025.101825

Simon Halvdansson

For time-frequency localization operators, related to the short-time Fourier transform, with symbol RΩ, we work out the exact large R eigenvalue behavior for rotationally invariant Ω and conjecture that the same relation holds for all scaled symbols RΩ as long as the window is the standard Gaussian. Specifically, we conjecture that the kth eigenvalue of the localization operator with symbol RΩ converges to

\frac{1}{2} erfc (\sqrt{2 π} \frac{k - R^{2} | Ω |}{R | \partial Ω |})

as R → ∞. To support the conjecture, we compute the eigenvalues of discrete frame multipliers with various symbols using LTFAT and find that they agree with the behavior of the conjecture to a large degree.

对于与符号RΩ的短时傅里叶变换相关的时频定位算子，我们计算出旋转不变量Ω的确切大R特征值行为，并推测只要窗口是标准高斯，相同的关系适用于所有缩放符号RΩ。具体来说，我们推测符号为RΩ的定位算子的第k个特征值收敛到12erfc（2πk−R2|Ω|R|∂Ω|）为R → ∞。为了支持该猜想，我们使用LTFAT计算了具有不同符号的离散帧乘法器的特征值，发现它们在很大程度上符合该猜想的行为。

引用次数: 0

Dynamical frames and hyperinvariant subspaces 动态框架与超不变子空间

IF 3.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Harmonic Analysis

Pub Date : 2025-11-13 DOI: 10.1016/j.acha.2025.101824

Victor Bailey , Deguang Han , Keri Kornelson , David Larson , Rui Liu

The theory of dynamical frames evolved from practical problems in dynamical sampling where the initial state of a vector needs to be recovered from the space-time samples of evolutions of the vector. This leads to the investigation of structured frames obtained from the orbits of evolution operators. One of the basic problems in dynamical frame theory is to determine the semigroup representations, which we will call central frame representations, whose frame generators are unique (up to equivalence). Recently, Christensen, Hasannasab, and Philipp proved that all frame representations of the semigroup

Z_{+}

have this property. Their proof of this result relies on the characterization of the structure of shift-invariant subspaces in

H^{2} (D)

due to Beurling. In this paper we settle the general uniqueness problem by presenting a characterization of central frame representations for any semigroup in terms of the co-hyperinvariant subspaces of the left regular representation of the semigroup. This result is not only consistent with the known result of Han-Larson in 2000 for group representation frames, but also proves that all the frame generators of a semigroup generated by any k-tuple

(A_{1}, \dots, A_{k})

of commuting bounded linear operators on a separable Hilbert space H are equivalent, a case where the structure of shift-invariant subspaces, or submodules, of the Hardy Space on polydisks

H^{2} (D^{k})

is still not completely characterized.

动态框架理论是从动态采样的实际问题中发展而来的，其中需要从矢量演化的时空样本中恢复矢量的初始状态。这导致了从演化算子的轨道中获得的结构框架的研究。动态框架理论的一个基本问题是确定半群表示，我们称之为中心框架表示，其框架生成器是唯一的（直到等价）。最近，Christensen， Hasannasab和Philipp证明了半群Z+的所有坐标系表示都具有这个性质。他们对这一结果的证明依赖于由于Beurling对H2(D)中平移不变子空间结构的表征。本文利用半群左正则表示的协超不变子空间，给出了任意半群的中心坐标系表示的刻画，解决了一般唯一性问题。这一结果不仅与Han-Larson（2000）关于群表示帧的已知结果相一致，而且证明了在可分Hilbert空间H上由交换有界线性算子的任意k元组（A1，…，Ak）所生成的半群的所有帧生成器都是等价的，而在多盘H2（Dk）上Hardy空间的移不变子空间或子模的结构仍未完全表征的情况下。

{"title":"Dynamical frames and hyperinvariant subspaces","authors":"Victor Bailey , Deguang Han , Keri Kornelson , David Larson , Rui Liu","doi":"10.1016/j.acha.2025.101824","DOIUrl":"10.1016/j.acha.2025.101824","url":null,"abstract":"<div><div>The theory of dynamical frames evolved from practical problems in dynamical sampling where the initial state of a vector needs to be recovered from the space-time samples of evolutions of the vector. This leads to the investigation of structured frames obtained from the orbits of evolution operators. One of the basic problems in dynamical frame theory is to determine the semigroup representations, which we will call <em>central frame representations</em>, whose frame generators are unique (up to equivalence). Recently, Christensen, Hasannasab, and Philipp proved that all frame representations of the semigroup <span><math><msub><mi>Z</mi><mo>+</mo></msub></math></span> have this property. Their proof of this result relies on the characterization of the structure of shift-invariant subspaces in <span><math><mrow><msup><mi>H</mi><mn>2</mn></msup><mrow><mo>(</mo><mi>D</mi><mo>)</mo></mrow></mrow></math></span> due to Beurling. In this paper we settle the general uniqueness problem by presenting a characterization of central frame representations for any semigroup in terms of the co-hyperinvariant subspaces of the left regular representation of the semigroup. This result is not only consistent with the known result of Han-Larson in 2000 for group representation frames, but also proves that all the frame generators of a semigroup generated by any <em>k</em>-tuple <span><math><mrow><mo>(</mo><msub><mi>A</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>A</mi><mi>k</mi></msub><mo>)</mo></mrow></math></span> of commuting bounded linear operators on a separable Hilbert space <em>H</em> are equivalent, a case where the structure of shift-invariant subspaces, or submodules, of the Hardy Space on polydisks <span><math><mrow><msup><mi>H</mi><mn>2</mn></msup><mrow><mo>(</mo><msup><mi>D</mi><mi>k</mi></msup><mo>)</mo></mrow></mrow></math></span> is still not completely characterized.</div></div>","PeriodicalId":55504,"journal":{"name":"Applied and Computational Harmonic Analysis","volume":"81 ","pages":"Article 101824"},"PeriodicalIF":3.2,"publicationDate":"2025-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145531480","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

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